Random space and plane curves

TL;DR

This paper investigates the properties of random knots generated by random trigonometric series, showing how the decay rate of coefficients influences knot complexity and revealing universal patterns in Alexander polynomial zeros.

Contribution

It introduces a model of random knots via random trigonometric functions and analyzes how coefficient decay affects knot types and properties, establishing conditions for tame knots.

Findings

Probability of complex knots decays as 1/N for certain coefficient decay rates.

Random tame knots correspond to functions with coefficient decay faster than k^{-3/2}.

Universal patterns observed in Alexander polynomial zeros and coefficients.

Abstract

We study random knots, which we define as a triple of random periodic functions (where a random function is a random trigonometric series, \[f(\theta) = \sum_{k=1}^\infty a_k \cos (k \theta) +b_k (\sin k \theta),\] with $a_k, b_k$ are independent gaussian random variables with mean $0$ and variance $\sigma(k)^2$ - our results will depend on the functional dependence of $\sigma$ on $k.$ In particular, we show that if $\sigma(k) = k^\alpha,$ with $\alpha < -3/2,$ then the probability of getting a knot type which admits a projection with $N$ crossings, decays at least as fast as $1/N.$ The constant $3/2$ is significant, because having $\alpha < -3/2$ is exactly the condition for $f(\theta)$ to be a $C^1$ function, so our class is precisely the class of random \emph{tame} knots. We also find some suprising experimental observations on the zeros of Alexander polynomials of random knots (with slowly and non-decaying coefficients), and even more surprising observations on their coefficients. Our observations persist in other models of random knots, making it likely that the results are universal.